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TRANSCRIPT
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An approach for solving two- person zero- sum matrix games in
neutrosophic environment
*1Khalifa bdelwahedAHamiden
Department, Institute of Statistical Studies and Research, Cairo University, Operations Research 1
Giza, Egypt 1Department of Mathematics, College of Arts and Science, Al- Badaya, Qassim University, Saudi
Arabia [email protected]
Abstract Neutrosophic set is considered as a generalized of crisp set, fuzzy set,
and intuitionistic fuzzy set for representing the uncertainty,
inconsistency, and incomplete knowledge about a real world problem. This paper aims to develop two-person zero- sum matrix games in a
single valued neutrosophic environment. A method for solving the game
problem with indeterminate and inconsistent information is proposed. Finally, two examples are given to illustrate the practically and the
efficiency of the method.
Keywords: Matrix games, payoff matrix, neutrosophic set, linear
programming, neutrosophic optimal strategy
1- Introduction Game theory is a mathematical modeling technique used for decision problems when there are two
or more decision makers in conflict or cooperation with each other. Each decision maker plays the
game to outsmart the others. Game theory provides many effective and efficient tools and techniques to mathematically formulate and solve many multi person ith strategies in tractions among multiple
rational DMs (Krishnaveni and Ganesan, 2018). Game theory is widely applied in many fields, such
as economic and management, social policy and international and national policies (Von Neumann and Morgenstern, 1944). Simple necessary and sufficient conditions for the comparison of
information structures in zero- sum games have been introduced by Peski (2008). The traditional
game theory assumes that all data of game are known exactly by players. However, there are some
games in which players are not able to evaluate exactly some data in our realistic situations. In these, the imprecision is due to inaccuracy of information and vague comprehension of situations by players.
For these, many researchers have made contribution and introduced some techniques for finding the
equilibrium strategies of these games (Berg and Engel, 1998), and Takahashi, 2008). In many scientific areas, such as system analysis and operations research, a model has to be set up
using data which is only approximately known. Fuzzy sets theory, introduced by Zadeh (1965), makes
this possible. Dubois and Prade (1980) extended the use of algebraic operations on real numbers to fuzzy numbers by the use a fuzzification principle. Bellman and Zadeh (1970) introduced the concept
f a maximizing decision making problem. Selvakumari and Lavanya (2015) and Thirucheran et al.
(2017) accelerated fuzzy game.
Based on the expected value operator and the trust measure of variables under roughness, Xu and Yao (2010) discussed a class of two- person zero- sum games payoffs represented as rough.
*Corresponding author
ISSN: 1735-8272, Copyright c 2019 JISE. All rights reserved
Journal of Industrial and Systems Engineering
Vol. 12, No. 2, pp. 186-198 Spring (April) 2019
- 187 -
Campos (1989) studied game problem under fuzziness in the goal and payoffs. Sakawa and Nishizaki (1994) used max- min principle of game theory to study single and multiobjective matrix
games with fuzzy goals and payoffs. Kumar (1988) investigated the max- min solution for the
multiobjective game problem. Bector et al. (2004a, 2004b) showed that a two- person zero- sum
matrix game with having fuzzy goals and fuzzy payoffs is equivalent to a pair of LPPs, each of them are the dual to the other in fuzzy sense. Vijay et al. (2005, 2007) based on the fuzzy duality; fuzzy
relation approach and ranking function for solving fuzzy matrix game. Pandey and Kumar (2010)
proposed a modified approach based on new order function for solving multiobjective matrix game with vague payoffs. Nan et al. (2010) studied fuzzy matrix game and a Lexicographic methodology
for finding the solution for it. Sahoo (2017) proposed a solution methodology for solving fuzzy matrix
game based on signed distance method. Li and Hong (2012) proposed an approach for solving constrained matrix games with triangular fuzzy numbers payoffs. Sahoo (2015) introduced a new
technique based on parametric representation of interval number to solve game problem.
Bandyopadhyay et al. (2013a) studied matrix game with triangular intuitionistic fuzzy number payoff.
Bandyopadhyay and Nayak (2013b) studied symmetric trapezoidal fuzzy number matrix game payoffs, where they transformed it into different lengths interval fuzzy numbers. Chen and Larboni
(2006) defined matrix game with triangular membership function and proved that two person zero-
sum game with fuzzy payoff matrices is equivalent to two linear programming problems. Seikh et al. (2013) proposed an alternative approach for solving matrix game.
In this paper, based on the concept of single valued trapezoidal neutrosophic numbers introduced
by Thamaraiselvi and Santhi (2016), we discuss a simple game, namely, the game in which the number of players is two and single valued trapezoidal neutrosophic payoffs which one player
receives are equal to single valued trapezoidal neutrosophic payoffs which the other player loses.
Here, linear programming problems and their implementation is being used in two- person zero- sum
matrix game theory. The outlay of the paper is organized as follows: In section 2; basic concepts and results related to
fuzzy numbers, trapezoidal fuzzy numbers, intuitionistic trapezoidal fuzzy numbers, and neutrosophic
set are recalled. In section 3, two- person zero sum game in neutrosophic environment is formulated. In section 4, an approach for solving matrix game is proposed. In section 5, two numerical examples
are given to illustrate the efficiency of the solution approach. Finally, some concluding remarks are
reported in section 6.
2- Preliminaries In order to discuss our problem conveniently, basic concepts and results related to fuzzy numbers,
trapezoidal fuzzy numbers, intuitionistic trapezoidal fuzzy numbers, and neutrosophic set are recalled.
Definition1. (Zadeh , 1965). A fuzzy set A~
defined on the set of real numbers R is said to be fuzzy
number, if its membership function 1,0:~ RA
has the following properties
(i) A~
is an upper semi- continuous membership function;
(ii) A~
is convex , i. e., yxxAAA~~~ ,min)1( , 1,0 , for all Ryx ,
.
(iii) A~
is normal, i. e., Rx 0 for which ;10~ xA
(iv) Supp 0:~
~ xRxAA
is the support of the A~
, and its closure cl (supp ))~
(A is
compact set.
Definition 2. (Kaufmann and Gupta, 1988). A fuzzy number 4321 ,,,~
aaaaA on R is a
trapezoidal fuzzy number if its membership function 1,0:~ RxA
has the following
characteristics
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Otherwise
axaaa
xa
axaaa
ax
xA
,0
,
,
43
34
4
21
12
1
~
Definition 3. (Atanassov , 1986). Let X be a nonempty set. An intuitionistic fuzzy set IA~
of X is
defined as XxxxxAII AAI :,,
~~~ , where x
IA~ and x
IA~ are membership and non-
membership function, respectively such that xIA
~ , xIA
~ : 1,0X and xIA
~0 +
1~ xIA
, for all .Xx
Definition 4. (Atanassov , 1986). An intuitionistic fuzzy subset XxxxxAII AAI :,,
~~~
of R is called an intuitionistic fuzzy number if the following conditions hold:
(i) There exists Rm such that 1~ mIA
, and ,0~ mIA
(ii) IA
~ is continuous function from 1,0R such that xIA
~0 + 1~ xIA
, for all
Xx , and
(iii) The membership and non-membership functions of IA~
are
;,0
),(
),(
,0
3
32
21
1
~
xa
axaxg
axaxf
ax
xIA
,,0
),(
),(
,0
3
32
21
1
~
xa
axaxg
axaxf
ax
xIA
Where, ggff ,,, are functions from 1,0R , f , and
g are strictly increasing functions, and
,g and f are strictly decreasing functions with ,1)()(0 xfxf
and .1)()(0 xgxg
Definition5. (Jianqiang and Zhong, 2009) A trapezoidal intuitionistic fuzzy number is denoted by
43214321 ,,,,,,,
~aaaaaaaaAIT , where
443211 aaaaaa , with membership and
non-membership functions are
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,,0
,
,
43
34
4
21
12
1
~
Otherwise
axaaa
xa
axaaa
ax
xITA
.,1
,
,
43
34
3
21
12
2
~
Otherwise
axaaa
ax
axaaa
xa
xITA
Definition6. (Smarandache, 1998). Let X be a nonempty set. A neutrosophic set is defined as
1,0,,,:,,, xWxVxUXxxWxVxUxA
NA
NA
NA
NA
NA
NA
N
, where
xUN
A
, xVN
A
, and xW NA
are truth, indeterminacy, and falsity membership functions,
respectively, and there is no restriction on the summation of them, so xUN
A
0 + xVN
A
+
3xWN
A
and 1,0 is nonstandard unit interval.
Definition7. (Wang et al. 2010). Let X be a nonempty set. The single valued neutrosophic set SVN
A of X is defined as
XxxWxVxUxA
NA
NA
N
SV
A
N
:,,, , where xUN
A
, xVN
A
, and 1,0xUN
A
for
each Xx and xUN
A
0 + xVN
A
+ 3xWN
A
.
Definition8. (Thamaraiselvi and Santhi , 2016). Let 1,0,, ~~~ aaau , and Raaaa 4321 ,,,
such that .4321 aaaa The single valued trapezoidal neutrosophic number
aaa
uaaaaa ,,;,,, 4321 is a special neutrosophic set on the real numbers R , whose truth,
indeterminacy, and falsity membership functions are
,,0
,
,
,
)(
43
34
4
32
21
12
1
otherwise
axaaa
xau
axau
axaaa
axu
x
a
a
a
a
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,,1
,
,
,
)(
43
34
43
32
21
12
12
otherwise
axaaa
xaax
axa
axaaa
axxa
xv
a
a
a
a
,,1
,
,
,
)(
43
34
43
32
21
12
12
otherwise
axaaa
xaax
axa
axaaa
axxa
x
a
a
a
a
Where, ,,aa
u and a are the maximum truth, minimum indeterminacy, and minimum falsity
membership degrees, respectively. A single valued trapezoidal neutrosophic number
aaa
uaaaaa ,,;,,, 4321 may be expressed an ill- defined quantity about a , which is
approximately equal to 32 ,aa .
Definition 9. (Thamaraiselvi and Santhi, 2016) Let aaa
uaaaaa ,,;,,, 4321 , and
bbb
ubabbbb ,,;,,, 4321 be two single valued trapezoidal neutrosophic (SVTRN) numbers
and 0c , then
(1) ,,,;,,,)( 44332211 bababauubababababa
(2) ,,,;,,,)( 14332241 bababauubababababa
(3)
,0,0,,,;,,,
0,0,,,;,,,
0,0,,,;,,,
4411223344
4414233241
4444332211
bauubabababa
bauubabababa
bauubabababa
ba
bababa
bababa
bababa
(4)
,0,0,,,;/,/,/,/
0,0,,,;/,/,/,/
0,0,,,;/,/,/,/
4441322314
4411223344
4414233241
bauubabababa
bauubabababa
bauubabababa
b
a
bababa
bababa
bababa
(5)
,0,,,;,,,
0,,,;,,,
1234
4321
cuacacacac
cuacacacacac
aaa
aaa
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(6) .0,,,;1
,1
,1
,1
1234
1
au
aaaaa
aaa
Definition10. Let aaa
uaaaaa ,,;,,, 4321 be a SVTRN number, then
(i) Score function ;)1()1(16/1)( 4321 aaavaaaaaS
(ii) Accuracy function )(aB )1()1(16/1 4321 aaavaaaa .
Definition11. Let ba, be any two SVTRN numbers, then
(i) If )()( aSaS then ba ,
(ii) If )()( aSaS , and if
(1) )()( bBaB , then ba ,
(2) (2) )()( bBaB then ba , and
(3) (3) )()( bBaB then ba .
3- Problem definition and solution concepts The two- person zero- sum game is the simplest case of game theory in which how much one
player receives is equal to how much the other loses. Parthasarathy and Raghavan (2010) studied the case when both players gave pure, mixed strategies. Nevertheless, the noncooperation between
players may be vague.
There are three types of two- person zero- sum single valued trapezoidal neutrosophic matrix games:
1. Two- person zero- sum matrix games with single valued trapezoidal neutrosophic goals, 2. Two- person zero- sum matrix games with single valued trapezoidal neutrosophic payoffs,
3. Two- person zero- sum matrix games with single valued trapezoidal neutrosophic goals and
single valued trapezoidal neutrosophic payoffs.
Consider a two player zero sum game in which the entries in the payoff matrix NIP are single
valued trapezoidal neutrosophic numbers. The neutrosophic pay-off matrix is
Player II
N
P =Player
N
mn
N
jm
N
m
N
m
N
ni
N
ji
N
i
N
i
N
n
N
j
NN
pppp
pppp
pppp
I
...
........
......
........
...
21
21
1...11211
(1)
Players ,I and II have n and m strategies, respectively denoted by M , and'M , respectively and are
defined as
m
i
ii
m xxRxM1
: 1,0: , (2)
and,
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.1,0:1
:'
n
j
jj
n yyRyM (3)
The mathematical expectation for player I is
ji
n
j
m
i
N
ji
N
yxpZ
1 1
, and for player II is
ji
m
i
n
j
N
ji
N
yxpZ
1 1
Where, N
ij
N
ij
N
ij ppp
N
ij
N
ij
N
ij
N
ij
N
ij uppppp ,,;,,, 4,3,2,1, .
Definition12. (Saddle point): If the min- max value equals to the max- min value then the game is
called a saddle point (or equilibrium) and the corresponding strategies are said optimum strategies.
The amount of payoff at an equilibrium point is the game value.
Remark 1. It is clear that the two mathematical expectations are the same since the sums are finite. Because of vagueness of payoffs single valued trapezoidal neutrosophic numbers, it is very difficult
for the players to choose the optimal strategy. So, we consider how to maximize player’s or minimize
the opponent's neutrosophic payoffs. Upon this idea, let us propose the maximum equilibrium strategy as in the following definition.
Definition13. In one two- person zero- sum game, player I 's mixed strategy
x player II 's mixed
Strategy y are said to be optimal neutrosophic strategies if yPxyPxyPx
NTTNT ~
for
any mixed strategies x and .y
Remark 2. The optimal neutrosophic strategy of player I is the strategy which maximizes N
Z
irrespective of II ' s strategy. Also, the optimal neutrosophic strategy of player II is the strategy
which minimizes N
Z irrespective of I ‘s strategy.
According to definition 10, and based on definition of the score value of each neutrosophic payoff N
ijp , the neutrosophic payoff matrix defined in (1) is reduced to the classical payoff matrix game as
Player II
P =Player
nmjmmm
niijii
nj
pppp
pppp
pppp
I
...
........
......
........
......
21
21
1.11211
(4)
Let us consider the game with deterministic payoff matrix (4), and the mixed strategies of players ,I
and II defined in (2) and (3), respectively. If is the optimum value of the game of a player II , then
the linear programming model for player II becomes
min
Subject to (5)
.,...,2,1,0;1
njyEyp jj
n
j
ij
Putting ,/' jj yy . Then problem (5) becomes
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mn
j
jj
y1
'max
Subject to (6)
jy
yp
j
j
mn
j
ij
,0
;1
'
'
1
Similarly, the linear programming model for player I is as
max
Subject to (7)
.,...,2,1,0
;1
mix
xp
i
i
nm
i
ij
Putting mixx ii ,...,2,1,/' . Then problem (7) becomes
m
i
ii
x1
'min
Subject to (8)
.;0
;1
'
'
1
ix
xp
i
i
nm
i
ij
4-Solution procedure In this section, a solution procedure for solving the two person zero sum matrix game problem is introduced as in the following steps:
Step1: Calculate the score value of each neutrosophic payoff N
ijp in (1) and obtain the classical
matrix game (4)
Step2: Translate the payoff matrix (4) into the corresponding problem (5) into problem (7)
Step3: Convert all of problems (5) and (6) into the corresponding linear programming problems (6),
and (8), respectively. Step4: Apply the simple method or any software ( Lingo) to solve problems (6),and (8), to
obtain the optimal strategies and the corresponding neutrosophic matrix game value for players ,II
and I , respectively.
5- Numerical examples Example 5-1- Solve the two- person zero- sum matrix game with trapezoidal neutrosophic numbers
4.,8.,4.;22,19,17,154.,5.,6.;8,6,5,36.,6.,3.;14,10,8,5
4.,5.,6.;8,6,5,35.,4.,6.;22,19,15,127.,4.,5.;16,14,11,9
06.,6.,3.;14,10,8,53.,5.,7.;6,3,1,0
33
N
ji
N
pP
- 194 -
The corresponding crisp payoff: Player II
653
274
031
33IPlayerpP ij
Referring to problem (6) , we get
'
3
'
2
'
1max yyy
Subject to:
.0,,
,1653
,1274
,103
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
yyy
yyy
yyy
yyy
II The optimal strategy of player .1Table
Game value Optimal strategy Variables
3.52412 0.8 '
1y
0 '
2y
0.2 '
3y
Referring to problem (8), we have
'
3
'
2
'
1min xxx
Subject to:
.0,,
,1620
,1573
,1341
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
xxx
xxx
xxx
xxx
Table2. The optimal strategy of player I
Game value Optimal strategy Variables
3.52412 0.0271 '
1x
0.4233 '
2x
0.5496 '
3x
Thus
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Table3. The optimal neutrosophic strategy of Example5.1
Game value IIPlayer IPlayer
7.0,8.0,3.0;79936.14,79924.11,5561.9,14894.7 8.0'
1 y
0'
2 y
02'
3 y
0271.0'
1 x
4233.0'
2 x
5496.0'
3 x
Example 5-2- Consider the following payoff matrix game in neutrosophic environment
5.,4.,6.;22,19,15,125.,3.,7.;45,32,30,254.,5.,6.;8,6,5,33.,5.,7.;6,3,1,0
6.,2.,8.;28,21,17,146.,2.,8.;25,20,13,117.,2.,9.;32,26,19,166.,2.,8.;30,22,18,15
4.,8.,4.;22,19,17,152.,3.,6.;15,11,8,46.,6.,3.;14,10,8,57.,4.,5.;16,14,11,9
43
N
ji
N
pP
Referring to the definition of score value the above payoff neutrosophic matrix game can be reduced
to the corresponding deterministic payoff as:
IIPlayer
71921
1091211
6534
43IPlayerpP ij
According problem (6) , we have
'
4
'
3
'
2
'
1max yyyy
Subject to:
.0,,,
,17192
,11091211
,16534
'
4
'
3
'
2
'
1
'
4
'
3
'
2
'
1
'
4
'
3
'
2
'
1
'
4
'
3
'
2
'
1
yyyy
yyyy
yyyy
yyyy
Using the simplex method (Phase2), we get
II The optimal strategy of player .4Table
Game value Optimal strategy Variables
9.769 0 '
1y
0 '
2y
0.231 '
3y
0.769 '
4y
Referring to problem (8), we have
- 196 -
'
3
'
2
'
1min xxx
Subject to:
.0,,;17106
,11995
,12123
,1114
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
'
3
'
2
'
1
xxxxxx
xxx
xxx
xxx
I The optimal strategy of player .5Table
Game value Optimal strategy
9.769 0'
1 x
923.0'
2 x
077.'
3 x
It is clear from the duality theorem that the optimal strategy of player I is:
077.0,923.0,0,, '
3
'
2
'
1 xxx ,
which are the coefficients slack variables in the final table of the simplex method .
Thus,
Table6. The optimal neutrosophic strategy of example5.2
Game value IIPlayer IPlayer
6.0,4.0,6.0;414184.27,024101.21,348888.16,526527.13 0'
1 y
0'
2 y
231.0'
3 y
769.0'
4 y
0'
1 x
923.0'
2 x
077.0'
3 x
6-Conclusions Neutrosophic sets are being more generalization of intuitionistic fuzzy set, which provide an
additionally to introduce the indeterminacy through the uncertainty. In this paper, we have considered a two- person zero- sum matrix games with single valued trapezoidal neutrosophic numbers. Firstly,
we have given the definition of the game with trapezoidal neutrosophic payoffs and then proposed an
equilibrium strategy. Secondly, we proposed a solution procedure. Lastly, two numerical examples
illustrated our research methods. We have discussed only one kind of games with uncertain payoffs. But of course, there are games with uncertain payoffs which will be take in our consideration the
future.
- 197 -
Acknowledgments The author would like to thank the referees for their insightful, valuable suggestions and comments,
which improved the quality of the paper.
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